Physics · Mechanics · Grade 7

Newton's laws

Why does the scales read more when a lift starts moving, and zero if the cable snaps?

The situation

You are standing on a scale in the glass lift of a tall building in Tashkent. The moment the lift sets off upwards, the scale reading suddenly rises, and just before it stops, the reading drops. Yet your mass has not changed — so what does the scale actually measure? If the lift cable snapped, what would the scale show, and what would save you?

Topic

Newton's laws. Scales in a lift: when does weight change

Key ideas

Newton's second law
Weight and g-force
Weightlessness

Formulas

m \vec{a} = \vec{N} + m \vec{g}

Newton's second law for the student: the normal reaction force of the scale $\vec{N}$ and the gravitational force $m \vec{g}$ .

N = m (g + a), R = \frac{N}{g} = m (1 + \frac{a}{g})

The force on the scale (weight); $a$ is the acceleration of the lift, positive upwards. A bathroom scale shows force in kilograms: read $R$ as the force divided by $g$ .

a = g (\frac{R}{m} - 1)

Finding the acceleration of the lift from the scale reading.

t_{1} = \frac{v}{a}, s_{1} = \frac{v^{2}}{2 a}

Duration and distance of the acceleration phase (the braking phase is exactly the same).

{\vec{F}}_{i n} = - m \vec{a}, T = (M + m) (g + a)

The inertial force in the frame of the cabin; the cable tension — the masses of the cabin ( $M$ ) and the student ( $m$ ) add up.

Theory

What does the scale measure?

A scale measures not mass but the force with which you press on it — your weight. By Newton's third law this force equals the normal reaction $N$ with which the scale pushes back on you. When the lift is at rest or moves at constant velocity, $a = 0$ and $N = m g$ — the scale shows your true mass.

Acceleration upwards — heavier

When the lift sets off upwards, or brakes while going down, the acceleration points upwards. To make you accelerate along with it, the scale has to push harder: $N = m (g + a) > m g$ — this is overload (extra apparent weight). If the acceleration points downwards (setting off downwards, stopping at the top), $N < m g$ and you feel lighter.

Weightlessness and the safety gear

If the cable snaps, the cabin and you fall freely together with acceleration $g$ : $a = - g$ , so $N = 0$ — the scale shows zero, which is weightlessness. Astronauts on a space station are also in constant free fall. In a real lift, the speed governor triggers the safety gear when the speed exceeds the rated speed by 15 %, and the safety gear clamps the cabin against the guide rails and stops it. In the model the average deceleration is $0.6 g$ (the standard is $0.2 g$ – $1 g$ ): during this, the scale shows $1.6 m g$ .

Model simplifications

The lift speed changes in a trapezoidal pattern: during the acceleration phases $a$ is constant, whereas real lifts increase the acceleration gradually for smoothness. The student moves together with the cabin as a single body, and the scale is ideal (no inertia). In free fall, friction with the air and with the rails is neglected. When the cable snaps, time is automatically slowed down by a factor of 4.

Inertial and non-inertial frames of reference

Newton's first law (the law of inertia): if no forces act on a body, or the forces balance, the body stays at rest or moves in a straight line at constant speed. Frames of reference in which this law holds are called inertial; a frame attached to the Earth is approximately inertial. A lift moving at constant speed is also inertial: everything in it happens just as it does at rest, and the scale shows $m$ — an experimental confirmation of the law of inertia. An accelerating cabin, on the other hand, is non-inertial: even if the student is at rest inside it, the forces $\vec{N}$ and $m \vec{g}$ acting on them are not balanced. To apply Newton's laws in such a frame, an inertial force ${\vec{F}}_{i n} = - m \vec{a}$ is added: $\vec{N} + m \vec{g} + {\vec{F}}_{i n} = 0$ . Switch on the “Cabin frame of reference” toggle — the inertial force is drawn instead of the velocity and acceleration. In free fall, ${\vec{F}}_{i n}$ exactly balances the gravitational force — weightlessness. Inertial forces in rotating frames of reference are covered in the “Non-inertial frames of reference” experiment (Foucault pendulum).

Mass, force and interaction

Mass is a measure of inertia: for the same acceleration, a heavier student needs a larger net force, $N - m g = m a$ — check the reading in “Measurements” with the mass slider. Mass is additive: the cable lifts the cabin and the student together, $T = (M + m) (g + a)$ . Third law: the student presses on the scale with force $\vec{P}$ , and the scale pushes back with force $\vec{N} = - \vec{P}$ ; two newton meters hooked together always show the same reading. Forces in mechanics: gravitational force (the “Law of universal gravitation” experiment), elastic force (the spring of the scale, the cable — “Elasticity and deformation”), friction (the safety gear clamps the rails). The gravitational force $m g$ does not depend on motion, but weight is the force exerted on the support — and it does depend on motion. The main task of dynamics: to find the motion from the forces, or the other way round; in “Mystery lift” you find the acceleration from the force on the scale. In a real lift the cabin is connected over a pulley to a counterweight — a system similar to the Atwood machine (the lab work in the “Inclined plane and Atwood machine” experiment).

Did you know?

In 1854, at an exhibition in New York, Elisha Otis stood on a freight platform himself and ordered the hoisting rope to be cut: thanks to his safety gear, the platform dropped only a few centimetres and stopped. Three years later, in 1857, the first passenger lift with such a safety gear went into service in New York.
On the International Space Station the force of gravity is only about 10 % weaker than at the Earth’s surface. The astronauts are weightless because they are in continuous free fall around the Earth together with the station.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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